Timeline for Construction of the petit Zariski topos out of the gros topos of a scheme
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2019 at 17:56 | answer | added | Ingo Blechschmidt | timeline score: 6 | |
| May 1, 2019 at 15:34 | comment | added | Ingo Blechschmidt | @Dean: The morphisms in $C$ are not what we might expect: Their ring-theoretic parts are required to be isomorphisms instead of local homomorphisms (as would be the case in the category of locally ringed spaces). The Spec functor is indeed an adjoint, if we let it map to the category of locally ringed toposes (objects are pairs $(\mathcal{E},\mathcal{O}_\mathcal{E})$, morphisms are pairs $(f:\mathcal{E}\to\mathcal{F}, f^\sharp:f^{-1}\mathcal{O}_\mathcal{F}\to\mathcal{O}_\mathcal{E})$), see for instance Sect. 12 of these notes. | |
| Apr 30, 2019 at 17:24 | comment | added | user30211 | The gros topos $E$ is the classifying topos for local rings. So maybe the $2$-category $C$ of toposes with maps into $E$ is like the category of locally ringed spaces. Then maybe $\text{Spec} : \text{Ring} \rightarrow C$ is adjoint to global sections. | |
| Nov 19, 2010 at 11:20 | comment | added | Marc Nieper-Wißkirchen | @Manny: Thanks for pointing out that discussion to me. Denis-Charles Cisinski's comments there were quite helpful to me. Furthermore, the discussion contains a link to a paper by Mathieu Anel, "Grothendieck topologies from unique factorisation systems" where my question is completely answered in the case that my scheme S is affine, i.e. Spec A: E classifies local rings in topoi while X classifies local rings that are localisations of A, so X is the subtopos of those objects Y of E such that A^1 restricted to Y is a localisation of A. This should generalise easily to general schemes S. | |
| Apr 19, 2010 at 9:22 | history | bounty ended | Marc Nieper-Wißkirchen | ||
| Apr 16, 2010 at 22:54 | comment | added | Manny Reyes | There was quite a long discussion on this topic over at the n-Category Cafe, here: golem.ph.utexas.edu/category/2009/01/… Have you seen it? | |
| Apr 12, 2010 at 20:30 | answer | added | Harry Gindi | timeline score: 2 | |
| Apr 12, 2010 at 8:29 | history | bounty started | Marc Nieper-Wißkirchen | ||
| Apr 7, 2010 at 11:43 | history | asked | Marc Nieper-Wißkirchen | CC BY-SA 2.5 |